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State Board · Class 8 · Mathematics

Lesson 8: Quadrilaterals: Constructions and Types

3 concepts · 24 practice questions with answers · Free to read

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Types of quadrilaterals and their properties

From daily life

A floor tile is a rectangle, a carrom board a square, a kite is kite-shaped, a door frame can be a parallelogram. All have four sides — but each has its own properties.

Simple definition

A closed figure with four sides is a quadrilateral. It has 4 angles and 2 diagonals. The four angles always add up to 360°.

In the figure: Family of quadrilaterals: parallelogram (blue) → rectangle (green) and rhombus (violet) → square (orange, with the properties of both); trapezium (grey) and kite (red) stand apart.

Live figure

Types of quadrilaterals and their properties

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  • Easy: All angles 90° and all four sides equal ⇒ square; all angles 90° but only opposite sides equal ⇒ rectangle.
  • Medium: Diagonals equal and bisecting each other at a right angle ⇒ square (in a rectangle the diagonals do not cross at a right angle).
  • Exam-type: A rhombus has diagonals 6 and 8: half-diagonals 3 and 4; side = √(9 + 16) = 5; perimeter = 4 × 5 = 20.

Properties

ShapeSidesAnglesDiagonals
Parallelogramopposite sides equal and parallelopposite angles equalbisect each other
Rectangleopposite sides equalall 90°equal
Rhombusall four sides equalopposite angles equalbisect at right angles
Squareall four sides equalall 90°equal and bisect at right angles
Trapeziumonly one pair parallel——
Kitetwo pairs of equal adjacent sides—cross at right angles
Remember

A square is both a rectangle and a rhombus; both of those are parallelograms.

Common mistake

Not every parallelogram is a rectangle. A rectangle needs 90° angles.

Shortcut

"Equal diagonals" = rectangle or square. "Perpendicular diagonals" = rhombus, square or kite. Memory clue: Rectangle — equal diagonals; rhombus — perpendicular diagonals; square — both.

Spot the difference

PointRectangleRhombus
Sidesopposite sides equalall four equal
Anglesall 90°opposite angles equal
Diagonalsequalbisect at right angles
In a squareboth sets of propertiesboth sets of properties

Practice questions

Easy
  1. 1Which quadrilateral has all angles 90° and all four sides equal?
  2. 2Which quadrilateral has only one pair of parallel sides?
  3. 3A rectangle is 9 cm long and 4 cm wide. Perimeter?
Medium
  1. 4A rhombus has side 6 cm. Perimeter?
  2. 5One angle of a parallelogram is 70°. The other three angles?
  3. 6A rhombus has diagonals 10 cm and 24 cm. Its side?
Challenging
  1. 7A rectangle has diagonal 13 cm and one side 5 cm. The other side and the perimeter?
  2. 8True or false: (a) Every square is a rectangle. (b) Every rectangle is a square.
Show answers (try them yourself first)
  1. 1A square.
  2. 2A trapezium.
  3. 32 × (9 + 4) = 26 cm.
  4. 44 × 6 = 24 cm.
  5. 5Opposite angle 70°; adjacent ones 180 − 70 = 110° and 110°.
  6. 6Half-diagonals 5 and 12; side = √(25 + 144) = √169 = 13 cm.
  7. 7Other side = √(169 − 25) = 12; perimeter = 2 × (5 + 12) = 34 cm.
  8. 8(a) True — every angle of a square is 90°. (b) False — adjacent sides of a rectangle can differ.

Angle sum 360° and angle problems

From daily life

Tear the four corners off a paper quadrilateral and join them at one point: they make a full circle, 360°.

Simple definition

Draw one diagonal and you get two triangles. Each has angle sum 180°, so the quadrilateral has 2 × 180° = 360°.

180° + 180° = 360°

In the figure: A diagonal makes two triangles: 180° + 180° = 360°; so the four angles add up to 360°.

Formula: Sum of 4 angles = 360° ; in a parallelogram adjacent angles = 180°

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Angle sum 360° and angle problems

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  • Easy: Three angles of a quadrilateral are 80°, 100°, 90°. The fourth is 360 − (80 + 100 + 90) = 360 − 270 = 90°.
  • Medium: In a parallelogram the adjacent angles are in the ratio 3 : 2. 3 + 2 = 5 ; 180 ÷ 5 = 36 ; angles 108° and 72°.
  • Exam-type: Angles of a quadrilateral are x°, (x + 20)°, (x + 40)°, (x + 60)°. 4x + 120 = 360 ; x = 60 ; angles 60°, 80°, 100°, 120°.
Remember

Cutting a quadrilateral into two triangles with a diagonal helps in many problems.

Common mistake

In a parallelogram opposite angles are equal and adjacent angles add to 180° — do not mix the two.

Shortcut

Given three angles, the fourth = 360° − their sum. Memory clue: A quadrilateral = two triangles, so 360° = 180° + 180°.

Spot the difference

PointParallelogramRectangle
Opposite anglesequalequal (90°)
Adjacent anglessum 180°each 90°
Sum of four angles360°360°
Example110°, 70°, 110°, 70°90°, 90°, 90°, 90°

Practice questions

Easy
  1. 1Three angles of a quadrilateral are 100°, 90°, 70°. The fourth?
  2. 2One angle of a parallelogram is 65°. Its adjacent angle?
  3. 3Four angles of a quadrilateral are 90°, 90°, 90° and x. x = ?
Medium
  1. 4The angles of a quadrilateral are x, 2x, 3x, 4x. Find each.
  2. 5Adjacent angles of a parallelogram are in the ratio 4 : 5. Find both.
  3. 6The angles of a quadrilateral are (x + 10)°, (x + 20)°, 2x° and (x + 30)°. Find each.
Challenging
  1. 7In trapezium ABCD, AB ∥ DC, ∠A = 70°, ∠B = 80°. Find ∠D and ∠C.
  2. 8In quadrilateral ABCD, ∠A = ∠C = 75° and ∠B = 2 × ∠D. Find ∠B and ∠D.
Show answers (try them yourself first)
  1. 1100 + 90 + 70 = 260; 360 − 260 = 100°.
  2. 2180 − 65 = 115°.
  3. 3360 − 270 = 90°.
  4. 410x = 360; x = 36; angles 36°, 72°, 108°, 144°.
  5. 54 + 5 = 9; 180 ÷ 9 = 20; angles 80° and 100°.
  6. 65x + 60 = 360; x = 60; angles 70°, 80°, 120°, 90°.
  7. 7Adjacent angles between parallel sides add to 180°; ∠D = 180 − 70 = 110°; ∠C = 180 − 80 = 100°.
  8. 8150 + 3∠D = 360; ∠D = 70°; ∠B = 140°.

Constructing quadrilaterals

From daily life

A carpenter making a table top must know which measurements are needed before cutting.

Simple definition

A general quadrilateral needs 5 independent measures (for example four sides and one diagonal). Special shapes need fewer.

5560°

In the figure: Rhombus KLMN: KL = 5, ∠K = 60°, KN = 5; arcs of radius 5 from L and N meet at M.

Table

ShapeMeasures neededExample
Square1side 5 cm
Rectangle2length 6, width 4
Rhombus2side 5 and one angle 60°
Parallelogram3two adjacent sides 6, 4 and angle 70° between them

In words: Steps (rhombus, side 5 cm, ∠K = 60°): 1) Draw KL = 5 cm. 2) Make a 60° angle at K. 3) Mark KN = 5 cm on it. 4) With radius 5 cm draw arcs from L and N; they meet at M. 5) Join LM and NM.

In words: Check: because the angle is 60°, triangle KLN is equilateral, so the diagonal LN measures 5 cm.

  • Easy: Rhombus: side 5 cm and ∠K = 60° — only 2 measures are needed.
  • Medium: Rectangle 8 × 6: 2 measures; after drawing, test the diagonal: √(64 + 36) = 10.
  • Exam-type: Quadrilateral ABCD: AB = 4, BC = 5, CD = 6, DA = 3 and diagonal AC = 6: 5 measures; ΔABC (4, 5, 6) and ΔACD (6, 6, 3) can both be drawn, so the quadrilateral can be drawn.
Remember

Five independent measures are needed to draw a quadrilateral; a square, rectangle or rhombus needs fewer.

Common mistake

Do not draw small arcs; they must be long enough to cross each other.

Shortcut

To finish a rhombus or parallelogram, find the last point as the meeting point of two arcs. Memory clue: Construction: first list the measurements, then draw a rough figure.

Spot the difference

PointRhombusParallelogram
Measures needed2 (side and angle)3 (two sides and angle)
Last pointby two equal arcsby two different arcs
Exampleside 5, ∠60°sides 6, 4, ∠70°
Checkdraw KM and test 5check opposite sides are equal

Practice questions

Easy
  1. 1A square has side 7 cm. Perimeter?
  2. 2Which quadrilaterals have equal diagonals?
  3. 3One angle of a rhombus is 65°. Find the other three.
Medium
  1. 4A rectangle has diagonal 17 cm and one side 8 cm. Find the other side.
  2. 5Opposite angles of a parallelogram are (2x + 10)° and (3x − 20)°. Find x and the angle.
  3. 6A rhombus has diagonals 30 cm and 16 cm. Find its side and perimeter.
Challenging
  1. 7Angles of a quadrilateral are in the ratio 2 : 3 : 3 : 4. Find each angle.
  2. 8In parallelogram ABCD, ∠A = 110°, AB = 6 cm, AD = 4 cm. Find the other angles and the perimeter.
Show answers (try them yourself first)
  1. 14 × 7 = 28 cm.
  2. 2Rectangle and square.
  3. 3Opposite angle 65°; adjacent angles 180° − 65° = 115° and 115°.
  4. 4Right triangle: 17² − 8² = 289 − 64 = 225 ; √225 = 15 cm.
  5. 5Opposite angles are equal: 2x + 10 = 3x − 20 ; x = 30 ; angle = 70°.
  6. 6Half-diagonals 15 and 8: side = √(15² + 8²) = √289 = 17 cm ; perimeter = 4 × 17 = 68 cm.
  7. 72 + 3 + 3 + 4 = 12 ; 360° ÷ 12 = 30° ; angles 60°, 90°, 90°, 120°.
  8. 8∠C = 110°, ∠B = ∠D = 180° − 110° = 70° ; perimeter = 2 × (6 + 4) = 20 cm.